LABORATORI NAZIONALI DI FRASCATI
SIS-Pubblicazioni
arXiv:hep-ph/9903331 v1 11 Mar 1999
LNF-99/006(P)
12 Febbraio 1999
IISc/CTS/4/99
hep-ph/9903331
Eikonalised minijet model predictions for cross-sections of photon
induced processes
Rohini M. Godbole1 , Giulia Pancheri2
1)
CTS, Indian Institute of Science, Bangalore, 5600 012, India
2)
INFN, Laboratori Nazionali di Frascati, P.O. Box 13, I-00044 Frascati, Italy
Abstract
In this talk I present the results of an analysis of total and inelastic hadronic cross-sections
for photon induced processes in the framework of an eikonalised minijet model(EMM).
We fix the various input parameters to the EMM calculations by using the data on photoproduction cross-sections and then make predictions for σ(γγ → hadrons) using same
values of the parameters. We then compare our predictions with the recent measurements
of σ(γγ
√ → hadrons) from LEP. We also show that in the framework of the EMM the rise
with s of the total/inelastic cross-sections will be faster for photon induced processes
than for the processes induced by hadrons like protons.
PACS13.60.Hb,13.85.Lg
Presented by the first author at the Workshop on photon interactions and the photon
structure, in Lund, September 10-13, 1998.
1 Introduction
The rise of hadronic total cross-sections σ(A + B → hadrons) with energy, has been
√
now observed for a set of comparable values of s where both A,B are hadrons [ 1, 2],
when one of them is a photon [ 3, 4] and when both of them are photons [ 5, 6]. It is well
known that interactions of a photon with another hadron or photon receive contributions
from the ‘structure’ of a photon which the photon develops due to its fluctuation into a
virtual q q̄ pair. The recent measurements [ 5, 6] of σ(γγ → hadrons) at higher energies
of upto and beyond ∼ O(100) GeV, have confirmed that the hadronic cross-sections rise
√
with s and preliminary claims [ 7, 8] are that they rise faster than the pp̄ cross-sections.
A measurement of γ ∗ p cross-sections by ZEUS collaboration, extrapolated to Q2 = 0 [
9, 10], lies above the photoproduction measurements [ 3, 4]. These extrapolated γ ∗ p data
√
and the new γγ data seem to indicate that the rise of cross-sections with s gets faster as
one replaces hadrons with photons successively. In the Pomeron-Regge picture [ 11] the
total cross-section is given by
tot
σab
= Yab s−η + Xab sǫ
(1)
where η and ǫ are related to the intercept at zero of the leading Regge trajectory and of the
Pomeron, respectively. The value of the Pomeron intercept indicated by the unpublished
results of ZEUS [ 9, 10] is 0.157 ± 0.019 ± 0.04 whereas the corresponding value for the
γγ data obtained by the L3 collaboration [ 5] is 0.158 ± 0.006 ± 0.028 which is to be
compared with the value of ∼ 0.08 [ 11] for pure hadronic cross-sections.
σtot(mb)
100
90
80
ZEUS γ proton (extr.) multiplied by 3/2*250
γ proton multiplied by 3/2*250
TPC, PLUTO, L3 γ γ multiplied by (3/2*250)**2
OPAL γ γ multiplied by (3/2*250)**2
70
60
50
40
proton-antiproton
proton-proton
30
20
10
10
2
10
3
10
√s ( GeV )
tot
Figure 1: Energy dependence of σab
.
2
4
Fig. 1 shows the energy dependence of the hadronic cross-sections as well as those
for the photon induced processes. The latter are multiplied by a quark model motivated
factor of 3/2 and the inverse of the probability for a photon to fluctuate into a q q̄ pair:
Pγhad . The value chosen for Pγhad is 1/250 which is close to a value motivated by VMD
picture. i.e.
X 4παem
1
Pγhad = PV M D =
≈
.
(2)
2
fV
250
V =ρ,ω,φ
Fig. 1 illustrates that all total cross-sections show a similar rise in energy when the difference between photon and hadron is taken into account, albeit with indications of somewhat steeper energy dependence for photon induced processes1 . The similarity in the
energy dependence makes it interesting to attempt to give a description of all three data
sets in the same theoretical framework[ 12]. We have analysed the cross-sections for
photon-induced processes alone [ 13, 14, 15] and we find that the EMM calculations
√
generically predict faster rise with s for γγ case than would be expected by an universal
pomeron hypothesis. I will further present arguments why in the framework of EMM
the Pomeron intercept is expected to increase as the number of colliding photons in the
process increase.
2 Eikonalised Minijet Model
There are some basic differences between the purely hadronic cross-section measurements and those of the total cross-sections in photon induced processes. In the purely
hadronic case the measurement of the total cross-section comes from the combined methods of extrapolation of the elastic diffraction peak and total event count, whereas in the
case of photon induced processes, this has to be extracted from the data using a Montecarlo; for example the total γγ cross-sections are extracted from a measurement of hadron
production in the untagged e+ e− interactions. Experimentally this gives rise to different
types of uncertainites in the measurement2 . Theoretically, the concept of the ‘elastic’
cross-section can not be well defined for the photons. In models satisfying unitarity like
those which use the eikonal formulation, it is important to understand what is the definition of the elastic cross-section and if the data include all of this elastic cross-section or
part or a small fraction. To that end let us summarise some of the basic features of the
Eikonal formulation.
√
It should be noted here that although the rate of rise with s is similar for both OPAL and L3 [ 7], on
this plot the OPAL data seems to stand a bit apart, which may be due to the difference in the normalisation
of the two data sets.
2
This is clear from the discussions, for example, in Ref. [ 7].
1
3
Let us start from the very beginning. Consider the eikonal formulation for the elastic
scattering amplitude
Z
ik
~
f (θ) =
d2~bei~q·b [1 − eiχ(b,s)/2 ]
(3)
2π
which, together with the optical theorem leads to the expression for the total cross-sections
σ
el
=
Z
σ tot = 2
d2~b|1 − e−iχ(b,s)/2 |2
Z
d2~b[1 − e−χI (b,s)/2 cos(χR )]
σ inel = σ tot − σ el =
Z
d2~b[1 − e−χI (b,s) ]
(4)
(5)
(6)
According to the minijet model[ 16, 17, 18, 19] the rise of total cross-sections can be
calculated from QCD. In the model, there is an ad hoc sharp division between a soft
component, which is of non-perturbative origin and for which the model is not able to
make theoretical predictions, and a hard component, which receives input from perturbative QCD. The minijet model assumes that the rise with energy of total cross-sections is
driven by the rise with energy of the number of low-x partons (gluons) responsible for
hadron collisions and in its simplest formulation reads
inel,u
σab
= σ0 +
Z
ptmin
d2 ~pt
jet
dσab
jet
= σ0 + σab
(s, ptmin ),
d2 p~t
(7)
the superscript u indicating that this is the ununitarised cross-section. This concept can
be embodied in a unitary formulation as in 5-6, by writing [ 20, 21]
inel
had
σab
= Pab
Z
d2~b[1 − e−n(b,s) ]
with
sof t
n(b, s) = Aab (b)[σh/a,h/b
+
jet
σab
(s, ptmin )
]
had
Pab
(8)
(9)
had
In eq.8, we have inserted, to include the generalization to photon processes, a factor Pab
defined as the probability that particles a and b behave like hadrons in the collision. This
2
parameter is unity for hadron-hadron processes, but of order αem or αem
for processes
sof t
with respectively one or two photons in the initial state. The definition of σh/a,h/b in eq.9
jet
had
is such that, even in the photonic case, it is of hadronic size, just like σab
(s, ptmin )/Pab
.
A simple way to understand the need for this factor [ 22] is to realise that the unitarisation
in this formalism is achieved by multiple parton interactions in a given scatter of hadrons
and once the photon has ‘hadronised’ itself, one should not be paying the price of Pγhad
for further multiparton scatters.
At high energies, the dominant term in the eikonal is the jet cross-section which is
calculable in QCD and depends on the parton densities in the colliding particles and ptmin ,
4
which admittedly is an ad hoc parameter separating the perturbative and nonperturbative
contributions to the eikonal. The basic assumption in arriving at eq. 8 is that the multiple
parton scatters responsible for the unitarisation are independent of each other at a given
value of b. In this model n(b, s) in eq. 9, is identified as the average number of collisions
√
at any given energy s and impact parameter b. The b dependence is assumed to be given
by the function Aab (b) which is modelled in different ways. This function measures the
overlap of the partons in the two hadrons a, b in the transverse plane.
Before going to the discussion of different models of Aab (b), we note that the minijet model is particularly well suited for generalisation to the photon-induced processes
where the concept of ‘elastic’ cross-section is not very well defined. Whereas for the
hadronic case one starts from the elastic amplitude followed by the optical theorem as
done in eqs. 3 – 6, in this case the starting point is actually the eq. 8 and then one
tot
defines σab
using eq. 6 with χR = 0 and using χI as given by 8. The above discussion
specifies the total cross-section formulation of the EMM for photon-induced processes.
While our earlier analyses [ 13, 14, 15] assumed that the γγ cross-sections presented
were the inelastic cross-sections, the analyis of [ 12] had used the total cross-section
formulation but with a different ansatz for the eikonal. Our analysis uses the total crosssection formulation of the EMM with the perturbative part of the Eikonal as given by
QCD a-la eq. 8.
3 Overlap function and jet cross-sections.
The overlap function Aab (b) is normally calculated in terms of the convolution of the
matter distributions ρa,b (~b)of the partons in the colliding hadrons in the transverse plane
Aab (b) =
Z
d2 b~′ ρa (~b′ )ρb (~b − ~b′ ).
(10)
If we assume that the ρ(~b) is given by Fourier Transform of the form factor of the hadron,
then Aab (b) is given by,
Aab (b) =
1
(2π)2
Z
~
d2~qFa (q)Fb (q)ei~q·b ,
(11)
where Fa,b are the electromagnetic form factors of the colliding hadrons. For protons this
is given by the dipole expression
Fprot(q) = [
ν2 2
] ,
q2 + ν 2
(12)
with ν 2 = 0.71 GeV2 . For photons a number of authors [ 20, 21], on the basis of Vector
Meson Dominance, have assumed the same functional form as for pion, i.e. the pole
5
expression
k02
Fpion (q) = 2
,
(13)
q + k02
with k0 = 0.735 GeV from the measured pion form factors, changing the value of the
scale parameter k0 , if necessary in order to fit the data.
Yet another philosophy would be to assume that the b-space distribution of partons
is the Fourier transform of the transverse momentum distribution of the colliding system [
23]. To leading order, this transverse momentum distribution can be entirely due to an
intrinsic transverse momentum of partons in the parent hadron. While the intrinsic transverse momentum (kT ) distribution of partons in a proton is normally taken to be Gaussian,
a choice which can be justified in QCD based models [ 24], in the case of photon the origin of all partons can, in principle, be traced back to the hard vertex γ q q̄. Therefore, also
in the intrinsic transverse momentum philosophy, one can expect the kT distribution of
photonic partons to be different from that of the partons in the proton. The expected functional dependence can be deduced using the origin of photonic partons from the γ → q q̄
splitting. For the photon one can argue that the intrinsic transverse momentum ansätze
would imply the use of a different value of the parameter k0 [ 15, 25], which is extracted
from data involving ‘resolved’ [ 26] photon interactions [ 27], in the pole expression for
the form factor. By varying k0 one can then explore various possibilities, i.e. the VMD/π
hypothesis if k0 = 0.735 GeV, or the intrinsic transverse distribution case for other values
of k0 [ 27].
The ansatz of eqs. 5– 6 and 9, requires that the overlap function be normalised to
unity, i.e.,
Z
d2~bAab (b) = 1.
(14)
Taking the form factor ansatz for the proton we then have
App (b) =
ν2
(νb)3 K3 (νb)
96π
and
Aγγ =
k03 b
K1 (k0 b)
4π
(15)
k02 ν 2
k02 ν 4
[K
(k
b)
−
K
(νb)]
+
νbK1 (νb)
(16)
0 0
0
2π(ν 2 − k02 )2
4π(k02 − ν 2 )
where ν and k0 are the scale factors mentioned earlier and Ki are the modified Bessel
functions.
had
If we look at eqs. 8- 9 it is easy to see that Aab (b) and Pab
always appear in the
had
combination Aab (b)/Pab [ 28, 26]. Hence only one of them can be varied independently.
Note also that σ soft can always be renormalised since it is a function fitted to the low
energy data. By looking at eq. 9 we can see that if the s-dependence of the jet crosssections were similar for all the colliding particles, then the difference in the s-dependece
Aγp (b) =
6
of the total/inelastic cross-sections can be estimated by looking at the behaviour of Aab . It
is also clear then that changing the scale parameter k0 in Aab (b) is equivalent to changing
had
Pab
. Note also that
had
had
Pγp
= Pγhad ; Pγγ
= (Pγhad )2 .
(17)
Hence, in analysing the photon-induced reactions, i.e, the γp and γγ cross-sections, the
only hadronisation probability that is an independent parameter is P had ≡ Pγhad .
Thus now we are ready to list the total number of inputs on which the EMM predictions depend:
soft
• The soft cross-sections σab
,
• ptmin and the parton densities in the colliding hadrons,
• P had and the ansatz as well as the scale parameters for the Aab (b),
Out of these the protonic and photonic parton densities are known from eP and eγ
soft
soft
DIS. The nonperturbative part σγp
, σγγ
has to be determined from some fits. We outline
the procedure used by us below. It is true that the jet cross-sections of eq. 7 depend very
strongly on the value of ptmin . Hence it would be useful to have an independent information of this parameter, which as said before separates the perturbative and nonperturbative
contribution in an ad hoc manner. Luckily, there is more direct evidence that the ansatz of
eq. 8 can describe some features of hadronic interactions. Event generators which have
built in multiple parton interactions in a given ab interaction, for the case of a, b being
p, p̄, were shown [ 29] to explain many features of the hadronic interactions such as multiplicity distributions with a ptmin around 1.5 GeV. Recent analyses of the γp interactions
seem to show [ 30] again that a consistent description of many features such as energy
flow, multiplicity distributions is possible with a ptmin value between 1.5 – 2.0 GeV.
jet
The energy dependence of σab
as defined in eq. 7 will of course get reflected in the
energy rise of the eikonalised total or inelastic cross-section. It is therefore instructive to
see how this depends on the type of the colliding particles. We compare this for pp̄, γp
and γγ case, where we have multiplied the γp and γγ jet cross-sections by factor of α
and α2 respectively. In the comparison of Fig. 2 we have used the GRV, LO parametrisations for both the proton [ 31] and the photon [ 32]. We note here that at high energies,
the rise with energy of the jet cross-sections is very similar in all the three cases, when
difference between a photon and a proton is accounted for. A study of the b-dependence
of the respective Aab (b) given by eqs. 15, 16, shows that the photon is much ‘smaller’
as compared to the proton in the transverse space, which is also understandable as the
photon after all owes its ‘structure’ to the hard γq q̄ vertex. Hence, we expect that the
7
10 3
10 2 QCD jet cross-sections with ptmin =2 GeV, GRV densities
σjets(mb)
10
1
10
10
10
10
10
γγ times 1/α
2
-1
-2
-3
γp times 1/α
-4
proton-proton
-5
1
10
√s ( GeV )
10
2
10
3
jet
Figure 2: Energy dependence of σab
for ptmin = 2GeV .
damping of the cross-section rise due to multiple scattering for photons will be less than
for a proton. This, coupled with the above observation of the jet cross-section, implies
that in the EMM, total/inelastic cross-sections are expected to rise faster with energy as
we replace a proton by a photon. I.e., an increase in the pomeron intercept as we go from
p̄p to γp and γγ as indicated by the data, is expected in the EMM framework.
4 Results
Now in this section let us spell out our strategy of fixing the various inputs to the EMM.
We restrict our analysis only to the photon-induced processes, i.e., γp and γγ crosssection. We follow the same procedure as we had adopted in [ 15], i.e., we fix all the
inputs to the EMM by a fit to the data on the available photoproduction data on σγp .
Here we do not include the data [ 9, 10], shown in Fig. 1, which has been obtained by an
soft
extrapolation of the low Q2 data to 0. We determine σγp
by a fit to the photoproduction
data using a form suggested in [ 21],
Aγp Bγp
soft
0
σγp
= σγp
+ √ +
.
s
s
(18)
0
We then determine Aγp , Bγp and σγp
from the best fit to the low-energy photoproduction
0
data, starting from the quark-model motivated ansatz σγp
= 2/3σp̄p . In earlier work [ 15]
we had used the inelastic formulation. Now we have repeated the same exercise with the
total cross-section formulation of the EMM, which we believe is the more appropriate to
use [ 12]. The results of our fit, using the total cross-section formulation of the EMM, are
8
10
-1
γp
σtot(mb)
ptmin=1.8,2,2.2 GeV total k0=0.66 GeV
Regge/Pomeron fit
jet cross-sections ptmin= 1.4, 1.6, 1.8, 2, 2.2, 2.4, 2.6 GeV
10
-2
1
10
10
2
10
3
√s ( GeV )
Figure 3: Comparison of the photoproduction data with the EMM fits with total crosssection formulation, and jet cross-sections as a function of ptmin .
shown in Fig. 3. The fit values of the parameters are
0
σγp
= 31.2 mb ; Aγp = 10 mb GeV;
Bγp = 37.9 mb GeV2 .
(19)
As compared with the similar exercise done in [ 15], we find that the rise of the eikonalised
√
cross-sections with s is faster in this case than in the inelastic formulation. However,
ptmin = 2 GeV is still the best value to use, as seen from Fig. 3. We use here the form
factor ansatz for the proton and the intrinsic kT ansatz for the photon with a value of
parameter k0 = 0.66 GeV, which corresponds to the central value from the measurement [
27] of the intrinsic kT distribution. We have used GRV distributions for both the proton
and photon and P had = 1/240. We also find, similar to the analysis in the inelastic
formulation by us [ 15] and others [ 21, 33], that the description of the photoproduction
data in terms of a single eikonal leaves leeway for improvement. We restrict ourselves to
the use of a single eikonal, so as to minimize our parameters but note that this can perhaps
be cured by using an energy dependent P had or alternatively an energy dependent k0 .
Now, having fixed all the inputs for the γp case, we determine the corresponding
parameters for the γγ case again by appealing to the Quark Model considerations and we
use,
2 sof t
sof t
.
(20)
σγγ
= σγp
3
All the other inputs are exactly the same as in the γp case. In this manner, we have really
extrapolated our results from γp case to the γγ case. The results of our extrapolation are
shown in Fig. 4.
9
1000
Pomeron/Regge band
Pomeron/SAS
900
mini-jet K0=0.66 (high),1(low) GeV inelastic EMM
800
mini-jet K0=0.66 GeV total EMM
700
ptmin=2 GeV GRV densities
500
400
γγ
σtotal(nb)
600
300
200
100
TPC
Desy 1984
DESY 86
LEP2-L3
LEP2-OPAL Preliminary
0
1
10
√s ( GeV )
10
2
10
3
Figure 4: EMM predictions for γγ cross-sections for total and inelastic cross-section
formulations along with data and Regge -Pomeron prediction.
Notice that the overall normalization of the photonic cross-section depends upon
P . When extrapolating from photoproduction to photon-photon using the inelastic
assumption, we had used P had = 1/200, which can be thought of as corresponding to
a 20% non-VMD component. Using the total cross-section formulation, the low energy
production data suggest rather to use P had = 1/240, a value which implies that the photon
is practically purely a vector meson. Then, Fig.4 shows that in the total cross-section
formulation, the extrapolation from γp to γγ leads to cross-section which lies lower at low
energies, but rises faster, than in the inelastic fits, for the same values of the parameters.
Both the inelastic and the total cross-section are seen to rise faster than is expected in
an universal pomeron picture. This feature is same both for γp and γγ cases. We show the
dependence of our results on the scale parameter k0 . The band in the figure corresponds
to using the Regge-Pomeron hypothesis of eq. 1, measured values of Xab , Yab for p̄p/pp,
γp case and the factorisation idea [ 34]
had
2
2
Xγγ = Xγp
/Xpp ; Yγγ = Yγp
/Ypp .
Here X(Y )pp stand for an average for pp and p̄p case.
We see that while our analysis using inelastic formulation and the default value of
k0 = 0.66 GeV [ 27] gave predictions closer to the OPAL data the total cross-section
formulation, for the same value of k0 , gives results closer to the L3 data, as already
pointed out in [ 12]. The sensitivity of the predictions to the difference between different parametrisations for the photonic partons increases with energy. At higher energies
one is sensitive to the low-x region about which not much is known. Our earlier analysis [
10
0.2
0.18
γγ with arbitrary normalization
0.16
γp in mb
0.14
σtot
0.12
0.1
ptmin=2 GeV, GRV densities, Phad=1/240
0.08
0.06
1
√s ( GeV )
10
10
2
Figure 5: Comparison of the energy dependence of the EMM predictions for the total γγ
and γp cross-sections with ptmin = 2 GeV, P had = 1/240.
15] in the inelastic formulation had shown that the γγ cross-sections rise more slowly for
the SAS [ 35] parametrisation of the photonic parton densities. The dependence of our
results in this analysis on the parton densities in the photon will be presented elsewhere [
36].
Fig. 5 shows a comparison of the eikonalised γp and γγ cross-sections, in the total
formulation, with the different parameters for γp and γγ case related as described before.
√
We see indeed that in the EMM the γγ cross-sections rise faster with s than γp case, as
was expected from the results shown in Fig. 3 and arguments following that. However,
the dependence of this observation on P had and/or the scale parameter needs to be still
explored.
5 Conclusions
In conclusion we discuss the results of an analysis of total and inelastic hadronic crosssections for photon induced processes in the framework of an eikonalised minijet model
(EMM). We have fixed various input parameters to the EMM calculations by using the
data on photoproduction cross-sections and then made predictions for σ(γγ → hadrons)
using same values of the parameters. We then compare our predictions with the recent
measurements of σ(γγ → hadrons) from LEP. We find that the total cross-section formu√
lation of the EMM predicts faster rise with s as compared to the inelastic one, for the
same value of ptmin and scale parameter k0 . In the former case our extrapolations yield
results closer to the L3 data whereas in the latter case they are closer to the OPAL results.
11
We also find that in the framework of EMM it is natural to expect a faster rise with
for the γγ case as compared to the γp case.
√
s
6 Acknowledgement
It is a pleasure to thank Goran Jarlskog and Torbjorn Sjöstrand for organising this workshop which provided such a pleasant atmosphere for very useful discussions. G.P. is
grateful to Martin Block for clarifying discussions on the total cross-section formulation.
R.M.G. wishes to acknowledge support from Department of Science and Technology (India) and National Science Foundation,under NSF-grant-Int-9602567 and G.P from the
EEC-TMR-00169.
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